VOLUME 112, NUMBER 13, APRIL 3, 2008
© Copyright 2008 by the American Chemical Society
ARTICLES QSPR Modeling of the Polarizability of Polyaromatic Hydrocarbons and Fullerenes Dana Martin, Sulev Sild, Uko Maran, and Mati Karelson* Institute of Chemistry, UniVersity of Tartu, 2 Jakobi Street, Tartu 51014, Estonia ReceiVed: October 16, 2007; In Final Form: January 11, 2008
Polarizability is one of the key properties determining the nonlinear optical effects of the new materials. In the current study, a quantitative structure-property relationship approach is used to model the polarizability of polyaromatic hydrocarbons (PAHs) and fullerenes. The model is derived using the data set of 40 PAHs and fullerenes and includes just one molecular descriptor, the AM1-calculated total molecular two-center exchange energy. The model is externally validated, and the obtained results are in good agreement with both the ab initio calculated and the experimental polarizabilities of these compounds. The reported quantitative structure-property relationship is a quick tool for finding an estimate of polarizability for different PAHs and fullerenes. Introduction The nonlinear optical (NLO) effects are widely studied due to the potential applications in telecommunications, computer storage, and optical devices.1 The materials that exhibit NLO activity can be characterized by the following features: high polarizability, asymmetric charge distribution, and acentric crystal packing. The NLO effects are large in materials with a high density of polarizable electrons. For instance, the organic compounds with extended π-electron systems are widely used in NLO applications.2 The fullerenes and the carbon nanotubes (CNT) are also characterized by an extended π-electron system and aromaticity,3 which make them and their derivatives potential candidates for NLO devices.4 The polyaromatic hydrocarbons (PAHs) are precursors in the synthesis of fullerenes and other carbon nanostructures5 and have aromaticity and extended π-electron systems.6-10 Therefore, the understanding and prediction of the polarizability of fullerenes and of their precursors may have significant technological importance. The polarization p induced in a molecule by a local electric field E can be expressed as an expansion in powers of the electric field (eq 1),11 * Corresponding author. E-mail: +3727375255. Fax: +3727375264.
[email protected].
Phone:
p ) µ01 + RIJEJ + βIJKEJEK + γIJKLEJEKEL + ...
(1)
where µ01 is the molecular dipole moment, RIJ is the linear polarizability (corresponding to the linear optical susceptibility at the macroscopic level), βIJK is the first hyperpolarizability (equivalent to the second-order nonlinear optical susceptibility at the macroscopic level), and γIJKL is the second-order hyperpolarizability (corresponding to the third-order nonlinear optical susceptibility at the macroscopic level). The linear dipole-dipole polarizability (R) controls the refraction index, whereas the second-order hyperpolarizability (γ) controls the intensity dependence of the refractive index.2,11 The photorefractive effect is the spatial modulation of refraction index due to the redistribution of charge in an optically nonlinear and conductive material when this is introduced in an electric field.12 The photorefractive effect is also the phenomenon that made possible the development of the optical storage media13 and optical switching devices.14 In order to estimate the suitability of a material for NLO devices, it is important to know the behavior of its refractive index; thus, knowledge about its polarizability and the second-order hyperpolarizability is important.
10.1021/jp7100368 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/12/2008
4786 J. Phys. Chem. C, Vol. 112, No. 13, 2008 The experimental polarizability of the bulk fullerenes can be obtained indirectly from the optical and conductivity measurements through the Clausius-Mossotti relation by using the experimentally measured dielectric constant, whereas the polarizability of the isolated C60 and C70 molecules can be measured by using the molecular beam deflection techniques.15,16 Different experimental techniques such as dielectric measurements, optical Kerr effect, and NMR studies in solutions17 have been used for obtaining the polarizability of PAHs. In the case of CNT, the third-order nonlinear optical susceptibility was experimentally investigated with Nd:YAG laser beams18 and by optical Kerr effect with a Ti:Sapphire laser.19 The theoretical calculation of the polarizability and hyperpolarizability for individual molecules is an important and helpful stage in selecting the substances with potential NLO behavior. One option to calculate NLO properties with high accuracy is using the ab initio methods with large basis sets and the inclusion of corrections for electron correlation. Because of the high computation expenses, these methods are not always easy to apply for systems with a large number of atoms such as PAHs and carbon nanostructures. Therefore, alternative schemes of calculation have been used. Such alternative computationally cheaper schemes for the calculation of polarizability are given in the following summary. Matsuzawa and Dixon20 have computed the polarizability and the second-order hyperpolarizability for a number of aromatic hydrocarbons and two fullerenes, C60 and C70, using a finitefield approach with the PM3 parametrization of the MNDO Hamiltonian. Using the Raman spectra, Snoke et al. 21 made a comparison of the bond polarizabilities for carbon-carbon bonds in hydrocarbons and fullerenes. They found that polarizabilities for single bonds in fullerenes and hydrocarbons compare well, whereas the double bonds in fullerenes have larger polarizability than in ethylene. On the basis of these polarizabilities a “bond polarizability” model was built, which sets the polarizability of the molecule equal to the sum of polarizabilities of the individual bonds. Hu and Ruckenstein22 constructed a bond polarizability model to predict the polarizability of fullerenes which respect the isolated pentagon rule (IPR). The model gives the formula R ) 1.262n to correlate the polarizability (R) of the fullerenes with the number (n) of carbon atoms. Moore et al.23 analyzed C60, C70, five isomers of C78, and two isomers of C84 and concluded that the static linear polarizability (R) is highly linearly correlated with the surface area. Ruiz et al.24 used a Pariser-Parr-Pople model, complemented with the criteria of physical consistency based on expected molecular response to a weak electric field, to predict the structure at the electronic level and the polarizability of five icosahedral fullerenes in the range of C60-C720. They found that the relationship between the polarizability (R) and the effective molecular radius (Re) is given by the R ) 0.75Re3 formula. Gueorguiev et al.25 investigated the size-dependent dielectric response of carbon fullerenes and found that the quantum size effects modify the size dependence of the polarizability to a significant degree. Applequist26 used the optimized atom monopole and dipole polarizabilities to fit the experimental mean polarizabilities and anisotropies of the alkanes and the planar PAHs. The atomic polarizabilities thus obtained were further used to calculate the polarizability tensors for other polycyclic aromatic compounds and fullerenes. The prediction of the model was compared with the polarizability of a conducting ellipsoid having dimensions close to those of the various fullerenes. The polarizability of
Martin et al. the ellipsoid is smaller in the case of a small fullerene but tends to converge for larger fullerenes.27 Mayer and co-workers28-30 presented a charge dipole model for calculating the polarizability of fullerenes and CNT, in which each atom is described by both a net electric charge and a dipole. The consideration of net charges enables one to address the fact that the π-electrons in metallic structures move from one atomic site to the other in response to an external field. Jensen et al.31 developed a dipole interaction model for calculating the polarizability of one- and two-dimensional clusters of C60. The model has been parametrized from the frequency-dependent molecular polarizability as obtained from the quantum chemical calculations for aliphatic, aromatic, and heterocyclic compounds. The same group32 has also developed a point-dipole interaction model to investigate the static polarizability and the second hyperpolarizability of the fullerenes and CNT and to study the dependence of these properties on the structure variation of the fullerenes and CNT. Brothers et al.33 deduced by using the density functional theory (DFT) a relationship between the longitudinal polarizability and the band gap of CNT. By applying this relation to the experimental band gap of different CNT their experimental polarizability can be predicted. The theoretically calculated molecular polarizabilities have been successfully used as the model descriptors in quantitative structure property relationship (QSPR) of different properties of chemical compounds.34-41 For the larger and more complex molecular systems, it has been possible to develop QSPR models for the polarizabilities themselves, based on more elementary molecular descriptors.42-44 In this work, the QSPR approach is applied to estimate the polarizability of PAHs and fullerenes using theoretically calculated molecular descriptors. For this, the forward selection of descriptors from large descriptor space is used. The proposed approach could provide a much less expensive alternative computation method for calculating properties of carbon nanostructures. Data and Methodology The polarizabilities of the following 18 PAHs (see Chart 1 for the structures of PAHs studied in this work) and 30 fullerenes were computed at the ab initio level using the Gaussian 03 program:45 acenapthene (ANP), anthracene (ATR), benzanthracene (BATR), 2,3-benzofluorene (23BF), benzo[a]pyrene (BP), biphenyl (BIPH), benzo(ghi)perylene (BGP), chrysene (CH), coronene (CO), dibenz[a,h] anthracene (DBA), fluoranthene (FA), fluorene (FL), naphthacene (NPTHC), naphthalene (NPTH), perylene (PRL), phenanthrene (PHN), pyrene (PY), triphenylene (TPHNL), C20(Ih), C28(Td), C32(D3), five isomers of C36 (C1, C2V, Cs, D2, D6h), C40(D2), five isomers of C44 (D2, C1, C2V, Cs, D3h), C50(D3), C60(Ih), C70(D5h), C72(D6h), C74(D3h), C76(Td), C78(C2V), seven isomers of C80 (two isomers with C2V symmetry, D2, D3, D5d, D5h, Ih), C82(C2), and C84(Td). The polarizability as calculated by the Gaussian program in au3 was transformed in Å3 to be in accordance with experimental polarizability and with other theoretical calculated polarizabilities found in the literature (see Tables 1 and 2). The structures were optimized at the DFT level using the B3LYP functional46 and the 6-31G* basis set.47 The polarizabilities of PAHs and fullerenes were calculated at the DFT level using the B3LYP functional and the 6-311G(d) basis set.48 This procedure has proved to be suitable for reproducing the experimental results.22 The data was separated in two sets. The compounds for which only the theoretical calculated polarizability is available were
QSPR for the Polarizability of PAHs and Fullerenes
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CHART 1: Structures of Polyaromatic Hydrocarbons
used for the training set, and compounds for which experimental polarizability was available were included into the external validation set. The training set (see Table 1) contains 12 PAHs and 28 fullerenes (ANP, BATR, 23BF, BP, BGP, CH, CO, DBA, FA, FL, NPTHC, TPHNL, C20(Ih), C28(Td), C32(D3), C36(C1), C36(C2V), C36(Cs), C36(D2), C36(D6h), C40(D2), C44(D2), C44(C1), C44(C2V), C44(Cs), C44(D3h), C50(D3), C72(D6h), C74(D3h), C76(Td), C78(C2V), two isomers of C80(C2V), C80(D2), C80(D3), C80(D5d), C80(D5h), C80(Ih), C82(C2), C84(Td)). The validation set (see Table 2) contains six PAHs and two fullerenes (ATR, BPH, NPTH, PRL, PHN, PY, C60(Ih), C70(D5h)). One of the purposes of this work was also to examine if the theoretical model obtained with the training set is good enough to reproduce the experimental polarizability of compounds in the validation set.
In order to develop the molecular descriptors for the studied compounds, the molecular structures of PAHs and fullerenes were first optimized at the semiempirical level using the AM1 parametrization49 with the MOPAC version 6.01 program.50 The calculations were carried out for the isolated molecules, and a 0.01 kcal/Å gradient norm was used. The CODESSA51 software package was subsequently used to obtain the molecular descriptors. The constitutional, topological, and geometrical descriptors were calculated using the optimized geometry of the molecule. The quantum chemical descriptors were calculated using the information extracted from MOPAC output files including the orbital energies and coefficients (and their combinations), atomic and bond populations, various components of the energy partitioning scheme, polarizabilities up to second order, and
4788 J. Phys. Chem. C, Vol. 112, No. 13, 2008
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TABLE 1: Ab Initio Calculated (pabinitio) and QSPR-Predicted Polarizabilities (pQSPR) of PAHs and Fullerenes of the Training Set, the % Error between pabinitio and pQSPR, and AM1-Calculated “Total Molecular Two-Center Exchange Energy” (Eexc(tot)) structure name
pabinitio, Å3
pQSPR, Å3
% errora
Eexc(tot) eV
ANP BATR 23BF BP BGP CH CO DBA FA FL NPTHC TPHNL C20(Ih) C28(Td) C32(D3) C36(C1) C36(C2V) C36(Cs) C36(D2) C36(D6h) C40(D2) C44(D2) C44(C1) C44(C2V) C44(Cs) C44(D3h) C50(D3) C72(D6h) C74(D3h) C76(Td) C78(C2V) C80(C2V) C80(D3) C80(D5d) C80(D5h) C80(Ih) C80(C2V1) C80(D2) C82(C2) C84(Td)
18.7 32.3 29.2 36.2 37.7 31.3 41.4 41.1 26.4 20.5 34.6 29.8 24.6 35.2 39.8 46.1 45.3 46.2 48.2 47.9 56.2 62.0 59.4 58.7 58.7 59.9 71.3 92.8 105.1 108.6 102.9 110.9 114.4 105.7 119.1 119.5 117.2 106.4 109.3 110.6
20.9 29.9 28.5 32.6 35.4 29.9 38.2 36.1 26.4 22.3 29.8 29.9 27.2 38.3 43.9 49.3 49.2 49.2 49.1 49.3 54.6 59.6 60.4 60.5 60.5 60.5 68.5 99.3 101.9 104.7 107.6 110.5 110.3 110.3 110.3 110.1 110.4 110.5 113.1 115.9
-11.2 7.9 2.6 10.3 6.3 4.6 8.2 12.9 0.2 -8.2 14.8 -0.2 -10.1 -8.2 -9.7 -6.7 -8.4 -6.4 -1.9 -2.9 2.9 4.0 -1.7 -2.9 -3.0 -1.0 4.0 -6.9 3.1 3.7 -4.4 0.3 3.7 -4.3 7.7 8.2 6.0 -3.9 -3.5 -4.7
-155.5 -224.7 -214.2 -246.2 -267.7 -224.8 -289.2 -272.9 -197.9 -166.0 -224.6 -224.7 -204.5 -289.8 -333.4 -375.4 -375.1 -374.8 -374.2 -375.8 -416.4 -455.0 -461.8 -462.0 -462.1 -462.6 -524.2 -763.5 -783.1 -805.2 -827.7 -849.1 -848.5 -848.5 -848.4 -846.9 -870.4 -850.3 -870.4 -891.8
a
% error ) (2(pabinitio - pQSPR)100)/(pabinitio - pQSPR).
dipole moments. In total, 328 descriptors were calculated. No experimental data were used as descriptors. The best QSPR model that correlates the polarizability calculated at the B3LYP/6-311G(d) level of theory and the molecular descriptors was derived using the heuristic method implemented in the CODESSA program.51 The heuristic method makes a preselection of the descriptors and builds the multidescriptor models taking into account the intercorrelation between descriptors, the Fischer criterion (F), and the squared correlation coefficient (R2) of the model. The QSPR model built for the training set was characterized by squared correlation coefficient (R2), leave one out cross validation coefficient (R2cv), square of standard error of multiple linear regression (s2), Fisher criterion (F), number of data points used for correlation (N), and number of descriptors used for QSPR model (n). For the compounds in the validation set, the polarizability values predicted by the QSPR model were compared with the experimental values,52-61 with the ab initio values calculated in the current paper, and with the polarizability calculated with other methods found in the literature20,21,23,25-27,62 (see Table 2).
Results and Discussion The best QSPR model, obtained for the ab initio polarizability of 12 PAHs and 28 fullerenes, is given by eq 2 and graphically presented in Figure 1. Importantly, a very good model was obtained with just one descriptor, the “total molecular two-center exchange energy” (Eexc(tot)) (t test ) -52.8930).
p ) 0.8715((1.3618) - 0.1290((0.0024)Eexc(tot) R2 ) 0.9863, R2cv ) 0.9845, s2 ) 16.2579, F ) 2797.67, N ) 40, n ) 1 (2) The molecular polarizability consists of an isotropic and an anisotropic part. The isotropic part of molecular polarizability is to a good extent an additive property63 depending on molecular size, indicating that polarizability can be calculated from a sum of transferable atomic or bond contributions. A proof that the polarizability is dependent on the number of atoms is the high correlation (R2 ) 0.9797, s2 ) 24.6879) between the polarizability and the number of C atoms or the polarizability and the number of atoms (R2 ) 0.9832, s2 ) 19.8977) in the molecule. The number of atoms refers to all atoms in molecule, in the present case to C and H atoms. The perfect additivity can only occur if the atoms are noninteracting, which is not the case for atoms in molecules. Miller64 underlined the need to take into account the atomic environment in calculating the polarizability of a molecule, and this is done with parameters assigned to atoms which characterize the hybridization of the atom. Consequently, the anisotropic part of molecular polarizability is largely influenced by the intramolecular interatomic interactions. The nature of the descriptor “total molecular twocenter exchange energy” (eq 3) more subtly allows the incorporation of anisotropic characteristics of a molecule in addition to isotropic characteristics.
Eexc(tot) )
∑Eexc(AB) ) ∑ ∑PµλPνσ〈µλ|νσ〉
(3)
The descriptor is a two-in-one measure considering both an isotropic and an anisotropic part. The anisotropic part is considered via the elements of the density matrix (PµλPνσ) that describe the overlap between atomic orbitals and consequently the interatomic interactions in molecules. The isotropic additive part is considered via the change in the number of electrons (or atoms) in different molecules. The use of Eexc(tot) as a descriptor slightly improves the squared correlation coefficient in comparison with classical additive descriptors (number of C atoms and number of atoms) and considerably improves the square of standard error of multiple linear regression (s2) in comparison with fully additive descriptors. Equation 2 was used to predict the polarizability for the compounds in the validation set. The correlation coefficient between the ab initio calculated values and the values predicted with linear regression model for the validation set is very good, R2 ) 0.9934 (graphically presented in Figure 1). The model does not apply well to estimate the polarizability of isomers of the same fullerene, which means that for a fine prediction of the polarizability in isomers other effects such as electron delocalization should be considered more precisely. In Table 2, the predicted polarizability with the linear regression model (eq 2) for the compounds in the validation set is provided. It can be observed that the values of the QSPRpredicted polarizability for NPTH, PHN, PY, BIPH, and C70 are very close to the average value of the experimental polarizabilities. The agreement between the values of predicted polarizability with the linear regression model and the experi-
QSPR for the Polarizability of PAHs and Fullerenes
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TABLE 2: Ab Initio Calculated (pabinitio) and QSPR-Predicted Polarizability (pQSPR), the % Error between pabinitio and pQSPR, Eexc(tot), the Polarizability Calculated (pcalcd) with Other Methods, and the Experimental Polarizability (pexpl) Data Found in Literature for Compounds in the Validation Set structure name
pabinitio, Å3
pQSPR, Å3
% error
Eexc(tot) eV
pcalcd, Å3
pexpl,Å3
NPTH ATR
15.6 24.3
17.4 23.6
-10.8 2.7
-128.4 -176.5
12.4, 17.01 19.5,a 25.49b
PRL PHN PY BIPH C60 C70
34.9 23.0 26.7 19.0 73.8 91.8
32.6 23.6 26.4 20.9 82.9 96.6
6.9 -2.9 1.2 -9.4 -11.6 -5.1
-246.2 -176.6 -198.0 -155.1 -635.9 -742.3
36.07b 24.76b 21.9,a 27.61b 21.51b 63.9,a 60.75,n 78,o 75.1,p 75.7q 73.76,n 89.8,u 88.3q
a
b
16.57, 17.40, 15.90, 16.47,f 17.40,g 16.75 ( 0.58 (av value) 25.40,h 30.56,i 26.10,i 27.73,i 25.33,c 25.41,j 25.97,d 26.77,e 30.05,k 27.04 ( 1.8 (av value) 39.5,i 43.0,l 41.25 ( 1.75 (av value) 23.53,c 24.94,d 21.17,e 23.21 ( 1.56 (av value) 28.23,c 24.80,e 26.52 ( 1.72 (av value) 19.62,m 20.77,d 20.19 ( 0.57 (av value) 76.5 ( 8,r 79.0,s 83.0,t 79.5 ( 2.66 (av value) 102 ( 14,V 97.0,s 103.5,t 100.83 ( 4.66 (av value) c
d
e
a
Finite-field approach, PM3 parametrization of the MNDO Hamiltonian, ref 20. b Calculated polarizability with the atom monopole-dipole interaction model, ref 26. c Cotton-Mouton effect in solution, ref 52. d Birefringence of crystals, ref 53. e Kerr effect in solution using Scolte theory, ref 54. f Kerr effect in solution, ref 55. g Stark spectrum of vapor, ref 56. h NMR in solution in electric field, ref 57. i Refraction, optical Kerr effect, and electrooptical absorption in solution, ref 58. j Optical Kerr effect in solution, ref 59. k Dielectric, Kerr, and NMR studies in solution, ref 17. l Electrooptical absorption on solution in static field, ref 60. m Kerr and Cotton-Mouton effects in solution at 589 nm, ref 61. n Atom monopole-dipole interaction (AMDI), ref 27. o Pariser-Parr-Pople approach, ref 24. p Ab initio, SCF/6-31++G, ref 62. q Bond order, bond polarizability model, ref 22. r Molecular beam deflection technique, ref 15. s Optical measurements: ellipsometry and reflection/transmission, ref 65. t Electron energy-loss spectroscopy, ref 66. u Ab initio, SCF/6-31++G, ref 68. V Gas-phase measurement; fullerene beam obtained with laser, ref 16.
energy” provides a useful tool in predicting the polarizability of PAHs and fullerenes. The model gives a rough estimation of the polarizability of PAHs and fullerenes, it is easy to compute, and it gives results comparable to those obtained with more demanding ab initio methods. The results obtained with the QSPR model also reproduce well the experimental values for compounds in the external validation set. Acknowledgment. EU Sixth Framework Program Project NANOQUANT (MRTN-CT-2003-506842) and Estonian Science Foundation (Grant Nos. 7709 and 7153) are acknowledged for the financial support. References and Notes
Figure 1. Relationship between the ab initio calculated and QSPRpredicted polarizabilities of PAHs and fullerenes.
mental polarizability for ATR and PRL is in acceptable limits. For C60, the experimental polarizability14,65,66 varies between 76.514 and 83.0 Å3.66 The polarizability calculated with the linear regression model, 82.9 Å3, is in this range of values. There is a relatively large difference between the different experimental polarizabilities for C60. One explanation is the different experimental methods by which these values have been obtained. With the exception of C60, for the other compounds in the training set there is an excellent agreement between the values predicted with the linear regression model and the ab initio calculated polarizability. The results obtained with the QSPR model are closer to ab initio and experimental values than those obtained at the semiempirical level20,67 or by atom monopole-dipole interaction for C60 and C70.27 Conclusions The QSPR model (R2 ) 0.9863) involving one descriptor, the AM1-calculated “total molecular two-center exchange
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